Dimension- and label-uniform stability of Gaussian propellers

Abstract

Assume the sharp finite Gaussian propeller inequality and its finite equality classification. We prove that the deficit in the squared first-moment functional controls the square of the Gaussian measure distance to a propeller, with one positive constant independent of the ambient dimension and the number of cells. Cells may have arbitrarily small measure, and no masses are prescribed. The proof passes to countable Gaussian score arrays, excludes infinite-support equality cases, and bounds the gain from all additional scores by the three-halves power of their total squared norm and the squared common mean of the leading triple. A quadratic estimate for that triple and a Gaussian margin bound then give the partition estimate. The square-root dependence of distance on deficit is sharp already for three planar sectors. The universal constant is positive but implicit.